Differentiability under equivalent norms
= Differentiability under equivalent norms
Suppose $f(a+h)=f(a)+Lh+r(h)$ and $|r(h)|/\|h\|_1\to0$. If $c\|h\|_1\le\|h\|_2\le C\|h\|_1$, then $|r(h)|/\|h\|_2\le c^{-1}|r(h)|/\|h\|_1\to0$. The boundedness of $L$ is also preserved, and the same argument in reverse proves equivalence of differentiability. The <derivative> is the same <linear map>, not a <norm>-dependent replacement.